Many algebro-geometric moduli spaces are constructed using Mumford's Geometric Invariant Theory (GIT) [1]}: given a reductive group \(G\)  acting linearly on a projective scheme \(Y\) , one obtains a quotient of an open semistable locus \(Y^{ss} \subset Y\) . Semistability is combinatorially determined via the Hilbert-Mumford criterion by calculating the weights of 1-parameters subgroups (1-PS) of \(G\) . In good cases, this enables a moduli-theoretic interpretation of GIT semistability from which one obtains a moduli space of semistable objects; the paradigmatic example being slope semistability for vector bundles on a smooth projective curve.
